Extreme points in the family of typically real polynomials

In this paper, we characterize the extreme points in the family of typically real polynomials of degree n or less. Viewed as a subset of the coefficient region which is a subset of n− 1 space, the set of extreme points is the union of two manifolds. Incase n is even, each of the two manifolds is determined by (n−2)/2 parameters that vary in [−1, 1]. If n is odd, then one of the manifolds is determined by(n−l)/2 parameters, and the other by (n −3)/2 parameters. We also determine sharp bounds for some of the coefficients.